This is a follow-up to my earlier post Anti-Racist Multicultural Math and I want to focus on some criticisms that Chris Correa is making about the Newton incident. He's disputing the charge that Anti-Racist, Multicultural Math is being taught in Newton, Mass. and believes that this story only has legs because of an agenda driven newspaper reporter and an audience only too willing to believe the worst about the educational establishment.

(Note: This turned out to be a much longer post than I envisioned . . . it's kind of taken on a "everything and the kitchen sink style" because at this point I'm too lazy to edit it, sorry, so the non-Newton stuff starts at the section entitled Know Thy Enemy.)

Let's review the facts:

- The downward trend in 6th grade math scores is unique to Newton and not seen statewide.
- The test is unchanged.
- The demographics are relatively unchanged.
- The SES of the students are relatively unchanged.
- The school district budget is at a record level despite fewer students in the district.
- Teacher turnover is low.
- Political & administrative oversight is unchanged.

The curriculm was changed to emphasize anti-racist core values and the results have been a steady decrease in 6th grade math achievement. Furthermore, in statewide rankings, the Newton School district has declined. In 2004, they ranked 39th in the state, in 2003 they ranked 25th, and in 2002 (the only ranking I could find) they ranked 7th. (See here for district rankings in 2003 and 2004.)Here is the raw data on the 6th grade Math scores:

GRADE 06 MATHEMATICS

PERFORMANCE LEVEL

2001

2002

2003

2004

ADVANCED

42

40

41

37

PROFICIENT

31

35

30

31

NEEDS IMPROVEMENT

16

17

19

21

WARNING

11

8

9

11

Now from what I gather, Chris is arguing that we should look at the current 8th grade math achievement scores because the 8th graders have 3 years of the anti-racist math under their belts and their scores are actually increasing:

GRADE 08 MATHEMATICS

PERFORMANCE LEVEL

2001

2002

2003

2004

ADVANCED

41

37

42

40

PROFICIENT

29

31

32

33

NEEDS IMPROVEMENT

20

21

15

20

WARNING

9

10

10

7

I'm not terribly reassured by an analysis of that data. Also, keep in mind that these current 8th graders were the 2002 class of 6th graders, the best performing cohort in the 4 year time window of that survey. Further, the 8th graders have had instruction under the anti-racist math curricula for only half of their school careers while the 6th graders have suffered for two thirds of their time in school. It's quite likely that the 8th graders were able to salvage some semblence of understanding of mathemetical fundementals in their first 4 years of school.
Chris also argues that the test scores are dropping because the proportion of disabled and limited English proficient students is increasing and that the difficulties of teaching students with these impairments is the root cause of the overall decline in 6th grade Math scores. I've taken the liberty of amalgamating the 2002, 2003, & 2004 data into the table below:

GRADE LEVEL 6 - MATHEMATICS

STUDENTS INCLUDED

PERCENT OF STUDENTS AT EACH PERFORMANCE LEVEL

#

%

A

P

NI

W/F

STUDENT STATUS - 2004

REGULAR

631

70

48

35

15

2

DISABLED

246

27

12

20

36

33

LIMITED ENGLISH PROFICIENT

26

3

19

46

27

8

STUDENT STATUS - 2003

REGULAR

658

71

52

34

13

1

DISABLED

237

26

11

23

36

30

LIMITED ENGLISH PROFICIENT

28

3

36

21

25

18

STUDENT STATUS - 2002

REGULAR

627

74

51

34

12

3

DISABLED

216

25

10

38

29

23

LIMITED ENGLISH PROFICIENT

9

1

-

-

-

-

Let's look at the trends in the 4 achievement categories. Notice that in the Advanced category, there has been an annual 1% rise in the number of disabled students who qualify while there has been a slightly downward trend for the regular students, and with only two data points for the limited English proficiency students we note a sharp drop off.

In the Proficient category, the regular students remain stable, the disabled students are declining in numbers and the limited English proficiency students are increading.

In the Needs Improvement category, the regular students,are showing a slow, but steady, increase, while 7% more disabled students have moved into this category over 3 years and the limited English proficient students have shown a marginal rise.

Lastly, in the Warning category, the regular students are pretty much holding steady, while 10% more disabled students have moved into this category and 10% fewer of the limited English proficiency students are classified in this lowest category.

Looking at the totality of these trends it's quite obvious that increasing the proportion of disabled and limited English proficiency by 4% doesn't explain the reallocation in performance amongst the groups of students. For Chris's hypothesis to hold, we'd expect to see the performance of regular students remain unchanged and most of the declines occuring amongst the disabled and limited English proficiency students. No such pattern is discernable.

Another explanation that Chris offers, but which I don't find exculpatory, is that none of the official Newton School District literature actually refers to "Anti-Racist Math." Further, if I'm reading him correctly, Chris believes that this whole story is the author's fabrication because the author quotes from a Social Studies Curricula Guide and that there is no anti-racist agenda within the Math curricula.

Between 1999 and 2001, under the direction of Superintendent Young and Assistant Superintendent Wyatt, the math curriculum was redesigned to emphasize "Newton's commitment to active anti-racist education" for the elementary and middle schools. This meant that no longer were division, multiplication, fractions and decimals the first priority for teaching math. For that matter, the teaching of math was no longer the first priority for math teachers, as indicated by the new curriculum guidelines, called benchmarks, which function as the primary instructional guide for teaching math in the Newton Public Schools.

Apparently, in looking at the Newton School District webpages, Chris was searching for explicit statements that could have served as the foundation for the above statements and he found none and apparently missed the significance of the statements on core values. Chris is looking at this from a functionalist point of view. By way of analogy consider a manufacturing company that makes quality a core value. No longer is quality strictly within the purvue of the quality control department, it has now become a mission for the entire company, and the first job of people on the assembly line is to ensure that their work meets the quality standards, and thereafter they can concentrate on productivity gains, workflow issues, labor relations, etc.

Effective multicultural education suggests a reexamination of the history, social constructs and dynamics related to race, class, gender, ethnicity, economics, and culture that impact curriculum and instruction. Multicultural education includes rigorous curriculum and inclusive teaching that challenges all students and staff. We are committed to developing a philosophy of multicultural education that can be infused across transformed curricula.
The district-wide Core Value--respect for human differences --set the direction and formed the basis for our continued support of the wide range of offerings above and for our espoused multicultural/anti-racist approach to teaching and learning.

Subsumed under these core values are specific functional objectives, and if these functional objectives are exempt from core values, we'd expect that to be noteworthy. The Mathematics Benchmarks read, in part, as follows:

The Newton Public Schools Mathematics Benchmarks for Elementary Grades (K-5) are organized to reflect the National Council of Teachers of Mathematics’ (NCTM) Principles and Standards for School Mathematics (2000). The Benchmarks include both content strands and process strands.

. . . These process strands are consistent with NCTM Principles and Standards, and with Newton’s commitment to active anti-racist education.

. . . However, the Benchmarks are not explicitly aligned with the 2000 revision of the Massachusetts Mathematics Curriculum Frameworks, as that document does not reflect all of the instructional and philosophical principles of NCTM or of the Newton Public Schools.

. . . In accordance with the core values of the Newton Public Schools, teachers are expected to provide a mathematics program which ensures that every child meets his or her grade level benchmarks, and that every child is challenged at his or her level.

This document is quite explicit in stating that the Newton District curriculm differs from the suggested state curricula by incorporating anti-racist principles developed locally and that the Math curricula is in accord with the dominant core values. I see no reason to conclude that the Math curricula is exempt from the core values of the District.

In the end, Chris asks us to not believe the worst about the educational establishment and asks us to accept the fact that the concept of Anti-Racist Math is something that simply doesn't exist.

KNOW THEY ENEMY

I however, disagree. In the spirit of knowing thy enemy, (Not Chris, but the advocates of fads like anti-racist math) I pull from my bookshelf, Sandra Harding's The Science Question in Feminism. Here are a few relevant quotes:

Page 36.

In the 1950s, the philosopher of science Willard Van Orman Quine indentified two dogmas of empriicism that he thought shoud be abandoned. "Modern empriricism has been conditioned in large part by two dogmas. One is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact, and truths which are synthetic, or grounded in fact. The other dogma is reductionism: the belief that each meanigful statement is equivalent to some logical construct upon terms which refer to immediate experience." Quine argued that both dogmas were illfounded, and that if they were abandoned, we would be inclined to see as less clear the purportedly firm distinction between natural science and specualtive metaphysics. We would also recognize pragmatic standards as the best we can have for judging the adequacy of scientific claims.

Page 39.

If we are not willing to try to see the favored intellectual structures and practices of science as cultural artifacts rather than as sacred commandments handed down to humanity at the birth of modern science, then it will be hard to understand how gender symbolism, the gendered social structure of science, and the masculine indentities and behaviors of individual scientists have left their marks on the problematics, concepts, theories, methods, interpretations, ethics, meanings, and goals of science.

Page 40

But we shall try to locate the pure, value-free core of science responsible for the purportedly inherent progressiveness in scientific method, in model claims in physics, in the mathematical language of science, and in logical reasoning. If, as I shall argue, pure science cannot be found in these places, then where should we try to find it?

Page 44.

I will argue that a critical and self-reflective social science should be the model for all science, and that if there are any special requirements for adequate explanations in physics, they are just that - special. (We will see that much of biology should already be conceptualized as social science . . . )

Page 45

Second, the concepts and hypotheses of physics require acts of social interpretation no less than do those in the social sciences. The social meanings that explanations in physics have for physicists and for the "man and woman in the street" are necessary components of these explanations, not scientifically irrelevant historical accidents. Perhaps it is appealing to imagine that the mathematical formulations of Newton's laws are the explanations of the movements of matter because it takes only a little effort for us modern folk to get a sense of what these formulas mean in ordinary language. But should we think of a formula so long that only a computer could read it in one hour as an explanation of a type of phenomenon? The answer to this question is "no." An explanation is a kind of social achievement. . .

The formula "1 + 1 = 2" is meaningless unless we are told what it is to count as a case of 1, of +, of =, and so on. . .

Scientific formulas are like legal judgements: the laws become meaningful only through learning (or deciding) how to apply them, and doing so is a process of social interpretation.

Page 47

I have been suggesting reasons for reevaluating the assumption that physics should be the paradigm of scientific knowledge-seeking. If physics out not to have this status, then feminists need not "prove" that Newton's laws of mechanics or Einstein's relativity theory are value-laden in order to make the case that the science we have is suffused with the consequences of gender symbolism, gender structures and gender identity.

Page 48

The belief that mathematics has no formal social dimensions - that the "external" social history of mathematics has left no traces on its "internal" intellectual structures - provides grounds for regarding science as fundamentally a set of sentences (such as Newton's laws) and physics as the paradigmatic science. . . . We have already argued that the explanations in physics cannot be "reduced" to mathematical "sentences" shorn of social interpretation.

Page 49

. . . two considerations make it plausible to regard as mythical the possibility of pure mathematics. In the first place, no conceptual system can provide the justificatory grounds for itself. To void vicious circularity, justificatory grounds must always be found outside the conceptual system one is trying to justify. The axioms of mathematics are no exception to this rule. . . . mathematical concepts and theories, too, are tested against historical social worlds they are designed to explain.

Page 50

They did so by replacing the social image of numbers as counting units with the social image of numbers as divisions of a line. These are social images because they reflect what people in historical cultures intentionally do. Not all cultures have been as preoccupied with the measuring - dividing a line - as has ours for the last few centuries. As one comentator points out, such a process of socially negotiating cultural images in mathematics is similar to what we do when we exclude patriotic killing in wartime from the moral and legal category of murder.

Page 51

It may be hard to imagine what gender practices could have influenced the acceptance of particular concepts in mathematics, but cases such as these show that the possibility cannot be ruled out a priori by the claim that the intellectual, logical content of mathematics is free of all social influence. . .

Mathematicians in this century, however, have found it impossible to justify the axioms of mathematics with any logical principles that are not more dubious, more counterintuitive, than the mathematics they are supposed to justify. So it is doubtful that the duty of providing a firm grounding for the truths of mathematics can be assigned to logic. Moreover, a few feminists have proposed ways in which specific assumptions in logic are androcentric. Merrill Hintikka and Jaakko Hintikka, for example, argue that the metaphysical units of a branch of logic called "formal semantics" correspond to masculine but not feminine ways of individuating objects.

OK, that's enough of that. The book goes on for 250 pages and if we substitute discriminatory racial ways of knowing for Harding's analysis of gender discrimation in our ways of knowing, we have an insight into the form that Anti-Racial Mathematics could take.

For us to expect that the Anti-Racist MultiCultural Math threat was simply the work of an agenda driven newspaper reporter, we'd have to discount the theoretical work on race, culture and gender coming from the academy, as well as the education establishment's well known penchant for implementing faddish approaches to education. Consider:

For Stacy Christ, a fourth grader in Fairfax County, Va., a homework problem about pencils and packages was an exercise in frustration.

"The answer required division," her mother, Susan, explained to me, "but she'd never been taught to multiply." . . .

But now "constructivism," as it is sometimes called, has become a force in teaching mathematics -- and the paradox is immense. In a field distinguished by reliance on proof, an unproven approach is being taken in thousands of schools.

The saga of whole math began in earnest in 1989, when the National Council of Teachers of Mathematics published standards that denounced a "longstanding preoccupation with computation and other traditional skills." According to the council, stressing addition, subtraction and, worst of all, memorization made students into "passive receivers of rules and procedures rather than active participants in creating knowledge."

The standards recommended that students get together with peers in cooperative learning groups to "construct" strategies for solving math problems, rather than sit in class with teachers instructing them.

Calculators were a necessity from kindergarten on, the council said, because students liberated from "computational algorithms" could pursue higher-order activities, like inventing personal methods of long division.

Dr. Frank Allen, a former council president and whole-math opponent, has noted that as the standards were being developed, the council's research advisory committee expressed concern about the failure of the standards commission to provide research support for its recommendations. But the standards' writers were undeterred, and today their views drive the direction of curriculums and textbooks in both public and private schools.

But DoEd's unqualified embrace of the constructivist approach--sometimes called the "New-New Math" -- prompted a counterattack by the heaviest artillery yet in the Math Wars. On November 18, 1999, Secretary Richard Riley and staff spilled their morning coffee over a full-page Washington Post advertisement signed by 200 mathematicians, scientists, and other experts calling on Riley to withdraw the federal endorsement of the 10 math programs. Among the signers were four Nobel laureates in physics and two winners of the Fields Medal, the highest honor for mathematicians.

The high-powered group protested the absence of active research mathematicians from DoEd's Expert Panel. They also objected that DoEd's Top-10 programs omitted basic skills, such as multiplying multi-digit numbers and dividing fractions.

"These programs [the Top 10] are among the worst in existence," said Cal State/Northridge math professor David Klein, who helped draft the letter. "It would be a joke except for the damaging effect it has on children."
Some of the panelists fought back. For example, Steven Leinwand accused the 200 scholars of being interested in "math for the elite" alone. Leinwand, math consultant for Connecticut's education department, said the NCTM and DoEd believe "math needs to empower all students." However, it was Leinwand who in 1994 wrote in Education Week that continuing to teach children multi-digit computational algorithms was "downright dangerous." . . .

Secretary Riley commented that NCTM has published "the prevailing standards in the country, so we thought that would make sense." But critics see a deliberate integration of ideological agendas. The architects of NCTM's 1989 standards declared that social injustices had given white males an advantage over women and minorities in math, and they promised NCTM's reinvented math would equalize scores. Equality would be achieved by eliminating the "computational gate."

Her son, who attends the public schools in New York City, was drawing clumps of sticks to solve multiplication problems in the 6th grade, she said.

In the 7th grade, he and his classmates were asked to find the area of a circle. Four weeks were devoted to the task. Traditionally, children were given the formula, but apparently these junior Archimedes were supposed to rediscover the uses of pi.

"Where's algebra?" Carson asked. "This is inquiry-based learning taken to an absurd end. But in schools of education, inquiry-based learning is IT!" . . . .

In both his prepared remarks and in discussion, Klein disparaged colleges of education, which produce most U.S. math teachers. In countries where students score high on standardized tests, teachers have degrees in mathematics, not math education, he said.

Klein said both parents and academic mathematicians are rebelling against the NSF and NCTM programs, which radically de-emphasize basic skill, encourage "rampant" calculator use beginning in Kindergarten, and falsely claim to teach conceptual understanding. "Instead they squander valuable class time on aimless projects with little or no intellectual content," he said. . . .

Carson said NCTM-style math education is closing the "performance gap" in test scores by lowering the achievement of above-average students.

How tame those struggles seem, however, when compared to the rising vanguard of self-described ethnomathematicians. For some, the new discipline just means studying the anthropology of various measurement methods; they merely want to supplement the accepted canon -- from Pythagoras to Euclid to Newton -- with mind-expanding explorations of mathematical ideas from other cultures. For others, however, ethnomathematics is an effort to supplant the tyranny of Western mathematical standards. . . .

''Mathematics is absolutely integrated with Western civilization, which conquered and dominated the entire world. The only possibility of building up a planetary civilization depends on restoring the dignity of the losers.''

'The practical effect,'' Klein says, ''has been watered-down math books that overemphasize inductive reasoning (like continuing visual patterns), because this is supposed to be good for women and minorities, and de-emphasizing deductive reasoning and mathematical proofs, which is the heart of mathematics, because that supposedly favors white males. . . .

''But mathematics is a worldwide monoculture. Look at the chalkboards in math departments at universities all around the world -- in Africa, Asia, Europe, Latin America. You will see the same symbols everywhere you go on this planet, except perhaps in colleges of education where fads reign supreme.'' Klein says he does spend some class time discussing the math of Mayans, Egyptians and other early civilizations. ''But ancient techniques and early discoveries in math will not take students very far who want to do something in the modern world with mathematics,'' he says. . . .

Rationale for introducing multicultural, anti-racist perspectives into the math curriculum, along with practical teaching ideas. Students address community issues through math. Includes sections on numerals, recording and calculating, geometry and measurement in architecture, geometry in art, data analysis, games of many cultures and more. Heinemann, 1996. 240 pp. * $25

Or see Anti-Racist Science Teaching by Gill & Levidow:

This book shows how science and technology embody distinctive values and cultural assumptions, including racist ones. These are in turn reflected in the way science is taught in many schools. Specific case studies present anti-racist approaches to biology, nutrition, and wildlife conservation, as well as one school's experiment with reorganizing the curriculum across disciplinary boundaries. Free Association Books (London), 1987. 324 pp. (01-609-5646/0507.)

Turning the focus back to the Newton District, it's likely that the administration changed the curriculm in order to close the achievement gap and they chose to use Connected Mathematics because of reports that:

Mathematics problem-solving scores for African American students in the two standards-based curricula were significantly higher than scores for African American students in the control group.

However, the administrators were unlikely to have investigated this fad too deeply for they would surely have come across this critique:

Putting aside the fact that the "Control" is neither pre-Standards nor alternatively reform, how good of a study is this? Well, of the 14,000 students in this district, one would assume over 1000 in Grade 6. This study involves 46 of them. Out of this group, 8 students were African American so the conclusion that the "African American students in the two standards-based curricula were significantly higher" is a comparison with performance from 8 students already in a Standards-based curriculum.

Further, if one analyses the performance of Newton's 6th grade African-American students, we actually see that the number of Advanced and Proficient students declines and the Needs Improvement and Warning students increases

The District of Newton's type of social experimentation and classroom indoctrination can lead to sad and devastating results. The first tremors of this quake are being felt in the lowered math performance of the 6th grade students, but as Joanne Jacobs has reported, elsewhere in the country lawsuits have been launched because diplomas are being withheld from students and the suits allege that this is due to culturally biased tests and even students with a 3.0 average can't pass the math portions of these tests:

With a 3.0 grade point average anchoring a solid academic record, Robyn Collins, 18, has big plans once she graduates from Reed High School in Sparks, Nev. . . .

The only problem is that she might not graduate from high school.

Collins is among 2,195 students -- 12 percent of the state's senior class -- who have completed all their course work requirements but will not receive high school diplomas this spring because they have not passed the math portion of Nevada's high school graduation test. . . .

"I've cried so much about this test," said Collins, who learned yesterday that she had failed the exam for at least the fifth time. "I'm not a stupid kid. . . . It is just that in my opinion, the stuff on the test doesn't equate to anything that I've learned in school."

Test proponents, however, contend that the exams are easy enough that the vast majority of students should pass them. They note that there are multiple opportunities to take the test and that many states offer remedial instruction, and they contend that high school graduates should demonstrate competence in basic skills.

Some states, in response to angry uproar about the damage to graduates' self-esteem, have lowered the pass threshold for tests, while others have implemented tests comprised strictly of pre-algebra material for their 10th grade students. Here are the Released Questions from the October 2004 test for California High School Exit Examination. God help us all if this test is too difficult for high school graduates to pass.

Considering the record of the educational establishment, their history of gullibility with respect to ill-considered fads, and academic research on anti-racist math and the availability of teacher resources to assist in eradicating evil racist mathematics from our schools, I find it entirely credible that an Anti-Racist Multicultural Math curriculm has been implemented and is not the figment of a newspaper reporter's imagination. The efforts Chris has made to offer alternate explanations are commendable but entirely unconvincing.

I got The Flight from Science and Reason with some Christmas money. Although its a fine collection of essays overall, there is a hilariously bad contribution from Mario Bunge, a philosophy professor at McGill.

It starts off good enough, attacking Heidegger, "radical feminist theory" and other brands of academic nonsense. Eleven pages into it, however, we find this:

Example 6: "Scientific Racism"

Racism is very old, but "scientific" racism is a 19th-century invention that culminated with the Nazi Rassenkunde and the accompanying extermination camps. The American version of this doctrine was introduced by some psychologists on the basis of flawed IQ measurements, and it was entrenched in the American legislation restricting immigration from Southern Europe and other regions. It was muted for a while in the wake of the revelation of the Nazi horrors, but it was resuscitated in 1969 by the Harvard professor Arthur Jensen, who, on the basis of some IQ measurements, asserted the innate intellectual inferiority of Afroamericans. This "finding" was unanimously rejected by the scientific community. In particular the Genetics Society of America warned against "the pitfalls of naive hereditarian assumptions"

Yaron Ezrahi, a member of the constructivist-relativist pseudosociology of science, claimed that this denial was due to ideological reasons. He held that the geneticists were particularly vehement in their criticisms of Jensen's work for being concerned, at least in part, with their own "public image and support." Ezrahi did not bother to analyze the very IQ tests from which Jensen had derived his "conclusions." Had he done so he might have learned that (a) such tests were indeed culture bound and thus likely to favor whites over blacks, and (b) no IQ test will be fully reliable unless it is backed up by a well-confirmed theory of intelligence--a theory that is lone overdue.

Yikes. Bunge couldn't even be bothered to get Jensen's institutional affiliation right!

From the point of view of the rest of society there's nothing wrong with "poor academic and economic performance" per se [since] the costs of such "poor" performance are borne by the actual "performers."

Though this debate started with Frank, I am not trying to single him out--this sort of argument seems common amongst loose and open borders libertarians, when they'll even admit that many immigrant groups aren't doing well economically or academically.

The problem with the "libertarian" argument in favor of loose/open borders is that, even in the absence of a welfare state (unlikely to ever happen), and the absence of government-funded infrastructure (even more unlikely to ever happen*), there are certain costs of poor-performing, high-crime groups that are unavoidable. Unless one plans to hole himself up on his own private property forever**, one will always have some exposure to his fellow citizens, including those who are poor, uneducated, and more likely to commit violent crimes. Thus, the only practical way for people to express their preferences as to the economic and academic characteristics of their new fellow citizens is through the government. I am certainly not questioning the idea of private property, but I *am* questioning the idea of abolishing public property. The average person's control over his circumstances should not be limited to a 1000 square foot house. A middle or even lower-middle class person should be able to choose to be insulated from crime and poverty without excessive cost. Under mass low-skill immigration, this is increasingly not an option; thus, one's freedom in a country that allows low-skill immigration is reduced, even free of welfare and infrastructure costs (which is, as I said before, highly unlikely to ever happen in an industrialized country).

*Even many small government conservatives and libertarians do not necessarily find such a situation desirable; in fact, I am not sure fully privatized transporation and education are good ideas. A private justice system is almost certainly a bad idea.
**Assuming one can afford a property far away from poverty and crime, which will become more difficult as the U.S. and Europe accept more poor/unskilled/uneducated immigrants

English people and Bantus have different frequencies of many genes. Suppose for simplicity that all English people are homozygous for a certain allele at a given locus, while all Bantus are homozygous for a different allele. An English man who marries an English woman will therefore have children with two ‘English’ genes each at the relevant locus. If on the other hand he marries a Bantu woman, his children will only have one ‘English’ gene each.

Therefore an English man who wishes to maximise the number and frequency of ‘English’ genes in the next generation should marry an English woman and not a Bantu.

Discuss.

The fallacy is that the argument considers only the offspring of the man, and not the other people affected by his choice. Let us assume for simplicity that there are equal numbers of English and Bantus, that all people marry, that all marriages are strictly monogamous, and that all couples have two children. (Relaxing these assumptions would just introduce irrelevant complications - the outcome would be the same, unless we introduce arbitrary assumptions about differential mating success of English and Bantus, or differential fitness of different mating combinations.)

On this basis, if an English man marries an English woman, 2 English people between them have 2 children with 2 ’English’ genes each. 2 English people therefore transmit in total 4 ’English’ genes to the next generation, giving 4/2 = 2 ‘English‘ genes transmitted per English person.

If on the other hand an English man marries a Bantu, we have to consider what happens to the English woman he would have married otherwise.

Suppose she marries a Bantu man, in which case 2 English people in total have 4 children with 1 ’English’ gene each. 2 English people therefore transmit in total 4 ’English’ genes to the next generation, giving 4/2 = 2 ‘English‘ genes transmitted per English person.

Alternatively she marries an English man, in which case 3 English people (one English couple and one English man married to a Bantu) in total have 4 children. 2 of the children have 1 ’English’ gene each, and 2 have 2 ‘English‘ genes each. 3 English people therefore transmit in total 6 ’English’ genes to the next generation, giving 6/3 = 2 ‘English‘ genes transmitted per English person.

Of course, if she does marry an English man, then some other English woman is left ‘spare’. Ultimately, given neutral assumptions, if an English man marries a Bantu woman, then an English woman somewhere along the line is constrained to marry a Bantu man, which gives us the case analysed in the last paragraph but one.

In each case the number of ’English’ genes transmitted per English person is the same. The number and frequency of English (and for that matter Bantu) genes in the population is therefore unaffected by the marital choices of the parents. Indeed, it hardly needs any examples to prove the point, since this is just a case of assortative mating, and assortative mating does not in itself affect gene frequencies in the population [Note 1].

It might seem unlikely that anyone would actually commit the fallacy described above, but Frank Salter, in his book On Genetic Interests (pp.260-264), certainly comes very close. In discussing marriage between different ethnic groups, he points out that partners from the same ethnic group will share some of the same distinctive genes, as a result of distant common ancestry, and that their children will therefore be more closely related to them than if they married someone from another ethnic group. Using the English/Bantu example, he calculates that endogamy (marriage within the group) increases kinship between parent and children by 92 percent as compared with exogamy. I have no strong disagreement with this part of the argument (though as I pointed out here, distant shared ancestry is not equivalent to recent ancestry for the purposes of kin selection). But Salter goes on to argue that ’choosing an English spouse over a Bantu one yields a fitness gain of 92 percent… The same applies in reverse order, so that a Bantu who chooses another Bantu instead of someone of English ethnicity has 92 percent more of his or her genes in offspring as a result. It is almost equivalent to having twice the number of children with an English spouse. Thus assortative mating can have large fitness benefits, the largest derived from choosing mates within geographic races.’

I think that anyone reading this, in the absence of any clear explanation to the contrary, would reasonably interpret Salter as concluding that interracial marriage reduces the aggregate fitness of the races concerned, or, what comes to the same thing, the overall frequency of their distinctive genes within the combined population. If it doesn’t, why worry? But for the reasons I have given, this conclusion would be quite fallacious. Either Salter has overlooked the fallacy, or he is aware of it but uses a misleading form of words. I think it is more likely that he has simply overlooked it, because he goes on to consider, in a confused way, the possible counter-advantages of exogamy, but nothing in this further discussion suggests he has grasped the point that population gene frequencies are not (in general) affected by assortative mating. He persists in assuming, fallaciously, that mating within an ethnic group in itself produces a fitness benefit. There is less excuse for the error because it is just another form of ‘Misunderstanding 10’ in Dawkins’s ’12 Misunderstandings of Kin Selection’, which I discussed here.

It may still be said, in a last-ditch defence of Salter, that an English man who marries a Bantu woman reduces his own aggregate ’genetic interest’, as defined in Salter’s theory, and therefore in some sense suffers a loss of fitness. If this does follow from the theory, so much the worse for the theory. But I am not sure that it does follow from the theory. ’Genetic interest‘, if interpreted consistently with Salter‘s other principles, can be either negative or positive. If you are positively related to someone in a given population, he increases your aggregate genetic interest in that population; if he is negatively related to you, he must therefore reduce it. In Salter’s own treatment of issues such as immigration, negative as well as positive relatedness is certainly taken into account: for example, he says ’it would appear to be more adaptive for an Englishman to risk life or property resisting the immigration of two Bantu immigrants to England than his taking the same risk to rescue one of his own children from drowning’ (p.67). By the same token, an English man enhances his genetic interest just as much by reducing the reproduction of pure Bantus as by increasing the reproduction of English people. Salter can’t have it both ways, counting negative interests when it suits him but ignoring them when it doesn’t.

Within the combined English-Bantu population, an English man is positively related to other English people, negatively related to Bantus, and has an intermediate relatedness to mixed English-Bantu people. If he marries a Bantu woman, he produces mixed offspring who are less closely related to him than English offspring would have been. (In my simplified example the relatedness is actually zero.) By marrying a Bantu woman he also constrains some English woman to marry a Bantu man, producing two more mixed offspring. But at the same time he constrains two Bantus (i.e. his own Bantu wife and the Bantu husband of the English woman) to marry English people, thus producing mixed rather than pure Bantu offspring. The overall reckoning is that four mixed offspring are produced instead of two pure English and two pure Bantu offspring. But the English man’s aggregate genetic interest is unaffected, since he is substituting one equivalent net outcome for another. The same conclusion holds, by a more technical route, if we use Salter’s own algebraic formulae. [Note 2] Salter’s fundamental error is that he considers only the immediate effect of a mating choice on the offspring of the person making the choice, while ignoring its repercussions elsewhere in the system. In particular, in considering the effect of marriage between an English man and a Bantu woman he fails to allow for the fitness benefit to the English man (from the point of view of his aggregate genetic interest) of preventing the birth of pure Bantu offspring. This must be an error even by Salter’s own principles.

I will return to the subject of Salter’s ‘ethnic genetic interests’ in general, but I thought it would be worth getting this more specific point out of the way first. It is quite an important one, as it invalidates most of what he says about inter-ethnic marriage.

Note 1: as usually defined, assortative mating is a tendency for individuals to mate more frequently with phenotypically similar individuals than would be expected in random mating. For example, Dictionary of Genetics, ed. King and Stansfield, says: ‘Assortative mating: sexual reproduction in which the pairing of male and female is not random, but involves a tendency for males of a particular kind to breed with females of a particular kind’. Mating may be either positively or negatively assortative (the latter tendency, ‘like mating with unlike‘, also being known as ’disassortative’ mating). Neither process in itself necessarily leads to any difference in mating success or the number of offspring, and the neutral assumption is that all phenotypes have the same average mating success. On this assumption any particular gene in any individual has the same probability of entering successful gametes, and the frequency of any given allele in the population is unaffected, though there will be departures from Hardy-Weinberg equilibrium in the proportions of homozygotes, and from linkage equilibrium between loci, if the genes concerned are implicated in the phenotypic resemblance . Of course, it is possible to imagine some circumstances in which assortative mating would affect mating success, for example, if one phenotype is very rare. Unless all phenotypes have the same frequency, it is also impossible to have perfectly disassortative mating without increasing the frequency of the rarer types. But there is nothing to suggest that Salter has such special cases in mind.

Note 2: we need to consider the effects of two alternative scenarios. In scenario A, an English man marries an English woman, and a Bantu man marries a Bantu woman. In scenario B, an English man marries a Bantu woman, and in consequence a Bantu man and an English woman somewhere along the line are constrained to marry each other. Using Salter’s formulae for relatedness (technically ’kinship’ rather than ’relatedness’, to be precise), in case A the English man has 2 offspring who are related to him by ¼ + ¾FST, while the Bantu couple have 2 offspring who are related to the English man by minus FST, where FST is the ‘coefficient of ethnic kinship‘ within each ethnic group as contrasted with the other. (See the Appendix to Salter’s book by Henry Harpending for an explanation and derivation of these formulae. Harpending shows that FST, which is usually a measure of genetic distance between populations, can also be interpreted as a measure of positive kinship between people in the same population, or negative kinship between people in different populations.) The overall effect of the first scenario is therefore to increase the English man’s aggregate genetic interest by 2(¼ + ¾FST) -2FST = ½(1 - FST). In case B, the English man and his Bantu wife have 2 offspring who are related to him by ¼ - ¼FST, while the Bantu man and the English woman (who we assume is unrelated to the English man except as a co-ethnic) have 2 offspring whose relatedness to the English man is ½FST - ½FST = 0. The overall effect of this scenario is to increase the English man’s aggregate genetic interest by 2(¼ - ¼FST) + 2(0) = ½(1 - FST). Thus the addition to the English man’s genetic interest is the same, i.e. ½(1 - FST), whichever choice he makes. In my simplified example, with English and Bantus 100% homozygous for different alleles, FST = 1, so the net effect on the English man’s genetic interest would be ½(1 - 1) = 0 for either option.

I read The Essential Difference by Simon Baron-Cohen on the plane recently. Weighting in at under 200 pages of relatively large font text, it's a quick read. Additionally, Baron-Cohen's narrative is skewed toward description and broad models rather than theoretical details, so there is a gentle learning curve when it comes to psycho-jargon.

According to the The Essential Difference there is an empathizing-systematizing spectrum in the human population. Baron-Cohen argues that males tend to be shifted toward systematizing, while females tend to be shifted toward empathizing. This is not a ground-shaking observation, and much of Baron-Cohen's argument seems to be formalizing and fleshing out what many would assume was "common sense."

One point that I did find interesting though: Baron-Cohen asserts that males are more adept at maintaining and scaling up dominance hierarchies. He makes this assertion based partially on a case study of a summer camp, though I have personally observed the same sort of tendency in secondary schools. Perhaps this is one reason that in general, there has been a shift toward more extreme patriarchy with the development of complex societies. "Complexity" is generally correlated with a profusion of the number of layers of formalized hierarchy that characterize a polity, social structures which males are better able to utilize. In contrast, females would likely flourish in a smaller scale society (the EEA), where interpersonal interactions loom large. This might explain the contrast in some modern societies like Japan, where the public domain is overwhelmingly male, but within the family women have a substantial amount of power and authority (stereotypically, I do not know if things have changed since the relative decline of Japan Inc.). When it comes to interpersonal dynamics, women often have the advantage, but when you scale up in numbers, female "empathizing" can not keep up with social complexity and the more narrow-focused and input insensitive male instincts are more effective. I would suggest that the drive toward feminism is partly an outgrowth of the democratization of Western society as established dominance hierarchies were overturned and rendered obsolete (I have argued before that "modern" liberal democracies might resemble the EEA more than the aristocratic or oligarchic societies that have been the norm for the past 5,000 years, in particular with the expansion of information technology which can transmit the faces, ergo, emotional states, of political leaders with high fidelity).

Two general predictions I will make based on this hypothetical model:

Larger secondary schools will tend to have a more androcentric social atmosphere, while smaller secondary schools will be typified by more parity.

Feminists who believe that centralized government power can be permanently leveraged to equalize relations between the sexes will be disappointed because of the nature of political organization and bureaucracy in these systems. Rather, the ultimate aims of feminists are likely more easily attained in the context of smaller political units, where the personal touch is more relevant because of a closer relationship between the leaders and their constituents.

A long article on both the scientific and personality dispute at the heart of the Flores "Hobbit" debate. One of the authors of the original paper notes that many of the opponents of the idea that the Flores find is a new species are multi-regionalists. Interestingly, the article mentions the presence of dwarfs in the local area (though there is no suggestion that they are microcephalics).

THE Pentagon considered developing a host of non-lethal chemical weapons that would disrupt discipline and morale among enemy troops, newly declassified documents reveal.

Most bizarre among the plans was one for the development of an "aphrodisiac" chemical weapon that would make enemy soldiers sexually irresistible to each other. Provoking widespread homosexual behaviour among troops would cause a "distasteful but completely non-lethal" blow to morale, the proposal says.

Other ideas included chemical weapons that attract swarms of enraged wasps or angry rats to troop positions, making them uninhabitable. Another was to develop a chemical that caused "severe and lasting halitosis", making it easy to identify guerrillas trying to blend in with civilians. There was also the idea of making troops' skin unbearably sensitive to sunlight.

The proposals, from the US Air Force Wright Laboratory in Dayton, Ohio, date from 1994. The lab sought Pentagon funding for research into what it called "harassing, annoying and 'bad guy'-identifying chemicals". The plans have been posted online by the Sunshine Project, an organisation that exposes research into chemical and biological weapons.

Spokesman Edward Hammond says it was not known if the proposed $7.5 million, six-year research plan was ever pursued.

Joanne Jacobs points me to an Anti-Racist Multicultural Math curriculm that isn't delivering the goods as well as expected. If you read my previous slam at Hispanic Math it shouldn't surprise you that I think this experiment is even further off the deep-end.

Newton, Massachusetts is the epicenter of this radical new approach to replace the old, stodgy racist math that we all know and love.

The school department was recently forced to publicly admit that the sixth-grade MCAS math scores have steadily declined over the past three years to the point where 32 percent of sixth-graders are now in the "warning" or "needs improvement" category. This means that if we were to attach a letter grade to these sixth-grade MCAS math results it would be a D-plus, with only 68 percent of the students passing. Brown Middle School fared so poorly that it is now subject to be placed under the federal No Child Left Behind Act for failing to keep pace under the minimum "adequate yearly progress guidelines."

Hmm, I wonder why they're having problems? Let's read on and seek answers to our questions.

The school department offered no tangible explanation for these declining scores other than to admit that they have no explanation, as articulated by Assistant Superintendent for Curriculum and Instruction Carolyn Wyatt (salary $106,804), "[The results] have decreased, incrementally, each year and continue to puzzle us." She went on to admit that this downward trend is peculiar to Newton and "is not being seen statewide." Again, she offered no explanation, but she did assure the School Committee that her assistant, Math Coordinator Mary Eich (salary $101,399), is currently investigating the problem.

Well, as long as the professionals are aware of the problem and investigating it, I think that's ok, don't you? Let's look at the likely causes:

But why have the sixth-grade MCAS scores plummeted in just three years? What mitigating circumstances, such as demographic or economic factors, could have contributed to this downward spiral?

Since Newton has been curiously alone in this decline, surely we can't blame the MCAS itself, especially since the test has hardly changed in just three years. The demographics of the city haven't shifted in so short a period. The socioeconomic level of the population has risen steadily. The school budget has dramatically increased - most notably with an unprecedented override in 2002 - to the point where the budget is at a record high, despite an actual decline in the number of students.

Class size has only recently increased, but mostly at the high schools and only sporadically at the lower grade levels. Since the turnover rate in the school department has always been low, the teachers and principals are roughly the same. We still have the same School Committee, superintendent and mayor.

So then, after eliminating any potential mitigating factors, what could possibly account for the steady decline in the sixth-grade math MCAS scores?

It is puzzling, isn't it?

The only logical and remaining explanation is change that occurred in the Newton math curriculum itself - the subject matter of what is taught and how, what is emphasized and what is not, what has been omitted and what is new. In short, what has changed in the elementary and middle school math curriculum to have affected such a dramatic decline in the MCAS scores?

Answer: the new math curriculum, otherwise known as anti-racist multicultural math.

Between 1999 and 2001, under the direction of Superintendent Young and Assistant Superintendent Wyatt, the math curriculum was redesigned to emphasize "Newton's commitment to active anti-racist education" for the elementary and middle schools. This meant that no longer were division, multiplication, fractions and decimals the first priority for teaching math. For that matter, the teaching of math was no longer the first priority for math teachers, as indicated by the new curriculum guidelines, called benchmarks, which function as the primary instructional guide for teaching math in the Newton Public Schools.

In 2001 Mr. Young, Mrs. Wyatt and an assortment of other well-paid school administrators, defined the new number-one priority for teaching mathematics, as documented in the curriculum benchmarks, "Respect for Human Differences - students will live out the system wide core of 'Respect for Human Differences' by demonstrating anti-racist/anti-bias behaviors." It continues, "Students will: Consistently analyze their experiences and the curriculum for bias and discrimination; Take effective anti-bias action when bias or discrimination is identified; Work with people of different backgrounds and tell how the experience affected them; Demonstrate how their membership in different groups has advantages and disadvantages that affect how they see the world and the way they are perceived by others..." It goes on and on.

These are the most important priorities that the school department has determined for teaching math from grade one through eight, as documented in the Newton Public Schools Benchmarks.

Nowhere among the first priorities for the math curriculum guidelines is the actual teaching of math. That's a distant second. To Superintendent Young and his School Committee, mathematical problem-solving is of secondary importance to anti-racist/anti-bias math.

Hey! I thought we had educational professionals in charge of curriculm development. How could something so assine have ever been implemented? Well, the logical error we've all made is that we've assumed that those seatwarmers occupying school district offices are actually professionals. See what damage can result from having the lowest IQ students finding their careers in the field of teaching and education administration.

As Joanne Jacobs points out: "If I wanted to keep poor, minority kids at the bottom of the social and educational ladder, I could find no better way than to devote class time to teaching them to see prejudice everywhere, rather than teaching them to add, multiply and find the common denominator."

As some of you may know, the google query Jared Diamond now has Guns, Germs and Gonads on page one. A reader points me to the fact that The Guardian is accepting questions, some of which Dr. Diamond will answer tomorrow (Feb. 13th) at 10 AM EST (registration necessary).

Over at Jinderella's blog a GNXP reader has suggested we do something like The Edge question (that is, what do you believe that you can't prove). Jin already posted one of my responses, but let me address my supposition from another angle....

Humans are a very stupid species on the cosmic "absolute" scale, even those on the right edge of the bell curve. There are innumerable truths right before our noses that we can not conceive of or discern because of cognitive limitations shaped by our evolutionary history.

Nevertheless, by the means of various mental aids (deductive logic, reproducible experiments, skepticism, etc.) we can glean the faint outlines of many of these truths in an indirect fashion. That is, data and analysis are not totally subjective, we are not totally trapped in our verbal/mental boxes.

I invite GNXP posters to offer their opinions in the space below as updates, while readers can react in the comment box.

I end by leaving you with a link to Robert Triver's response. If you have read that link, go browse Greg Cochran's comments about doctors.

jinn: Oh, why do I always have to go first? Well, I have thought about it-- I believe there is a biological basis for all behavior, and it is the genomic code married with its interactions with the phenome and the environment. And I believe we can understand and map the mechanisms completely. Eventually.

David Boxenhorn: A long time ago I realized that I had a fundamental belief that wasn't going to go away even if reason told me that the evidence points in the other direction: Life has meaning. My choice was thus: (1) Embrace my true belief and run with it where it would take me, (2) Deny my true belief and be depressed, (3) Neither embrace it nor deny it, and live a life of timid anxiety. I chose #1, and it has taken me quite far from my birthplace. Not long ago I coined the word 'logoism' to describe this belief, after a long search in which I turned up nothing, surprising me because I wanted nothing more than an antonym to nihilism. The lack of this term indicates to me that not enough people are thinking about it. What are the implications of meaning? God? Or could it be something else? More here.

I've wasted enough time on this (a few semi-grammatical comments over at Will's blog), it seems that these arguments when involving people with college educations devolve into working-back-to-first-principles and conscious or unconscious strawmen on both sides (because so many assumptions are left implicit and unelaborated). It seems clear that some signals of masculinity are contingent upon fashion. For example, during the Roman Empire, the preference for facial hair seemed to track the preference of particular emperors, that is, the tendency toward clean-shavenness of the early Empire gave way to beards in imitation of philo-Hellene Hadrian, while in the late 3rd century beards disappeared again, so much so that Julian the Apostate was mocked for his facial hair in the 4th century (again, in imitation of Greek philosophers). On the other hand, there are some male tendencies that seem invariant culture to culture. To be trivial, men seem to be more physically aggressive and less verbal. Additionally, I know of no culture where men and women are on average the same size, unlike say among gibbons where there is size parity, or among most whales, where females are larger. The size difference suggests some obvious implications when viewed through the lens of sexual selection.

Science does not always conform to our intuitions or our ideologies. The intersection between human sociology, psychology and biology is difficult to encapsulate in a few setences. Perhaps this is one reason why rejecting or muddling the salience of the above intersection is the preference of some.

Update below:

In Will's follow up post one commenter brings up what I call the men in pink point. That is, x was once considered masculine and !x was once considered feminine, but today the situation is reversed, ergo, masculine and feminine really don't mean anything.

The problem I have is that the data points to me all seem rather superficial and faddish, for example, the way men or women wore their hair, or the color associated with each gender in a culture. To me, it is like saying that there is no human universal social reaction to death because in Asia white is the color associated with mourning while in the West the color is black. My understanding is that the modern subdued gray/black formal uniform of men is derived from the defeat of the Cavalier cultural style by the Puritan cultural style. This is a historical outcome, and I don't see it as inevitable, and there are still cultures where males do all the "display."

The fact that I think is salient is one should reflect on whether during the 17th & 18th centuries, during the peak of Cavalier and Puritan cultural rivalry, did the relationship between males & females reflect the stylistic dichotomy??? I don't think so, by feminist lights I suspect both cultures would be judged "patriarchal."

It is true that there are great variances in the way males and females manifest their differences. In some West African cultures males are far less the typical "breadwinner" than is the case in the Middle East. Yet despite the fact that women have more public roles in West African cultures than in the Middle East, the vast majority of political-military leaders in both cultural traditions remained male. In both cultures I suspect women would put a priority on their role as mothers and evince very similar emotional attitudes toward threats to their children. There are basic differences undernearth the variation, contingent upon our evolutionary history in the EEA.

I make an ostentatious attempt on this blog to not be particularly topical in relation to "current events," but, in light of the recent post-tsunami news tsunami, I thought some readers might find some information on the Toba Event interesting. This mega-explosion in central Sumatra ~70,000 years ago might have ushered in a global winter, which palaeoanthroplogist Stanely Ambrose suggests precipitated a series of local population bottlenecks (more here). Ambrose's thesis basically points to Toba as the seminal event which shaped the present forms of various subpopulations which emerged out of the bottlenecks with a varied suite of specialized adaptations and features resulting from founder effect (a form of adaptive landscape theory it seems). The Real Eve, by Stephen Oppenheimer, also features the Toba Event as the watershed that separated the populations of East and West Eurasia into two clades.

A few months ago I posted about the possibility of a new great ape species. This report in Time convinces me that the scientist who was feeding the press most of the sensationalistic anecdotes is probably wrong that it is a new species.

Tucker Max runs what has to be one of the top five funniest sites on the net. His tales of debauchery are legendary -- he has a book and an MTV spot devoted to them. But it was nice to see some of his recommended books:

The Moral Animal, Robert Wright: A great intro to evolutionary psychology for the non-scientist. The beginning is boring, but after Chapter One it gets really good.

The Adapted Mind, Cosmides and To[o]by: The defining book on evolutionary psychology. This is where it all began, but it takes some background before you can really digest their writing.

Why we get sick, Neese and Williams. The best book you will ever read on medicine.

Most of Stephen Pinker, Mark Ridley, and Richard Dawkin's work, especially The Red Queen, The Selfish Gene, and The Language Instinct

Eibl-Eibesfeldt approaches the subject from a biological point of view, and includes evidence from other primates, from child psychology, etc, whereas Keeley looks at the archaeological and anthropological evidence. If you want to read just one book, I would recommend Keeley, as it is more up-to-date and has more detailed information on actual cases.

Both authors argue that in recent decades the prevalence and importance of war among primitive peoples has been understated by anthropologists. Eibl-Eibesfeldt comments that ‘the often repeated claim that hunters and food-gatherers in general are more peaceful than people at a higher cultural level is certainly false, and so obviously false that one wonders how it can persist so stubbornly in the face of the overwhelming abundance of long known facts’. Keeley concludes that ‘peaceful prestate societies were very rare; warfare between them was very frequent, and most adult men in such groups saw combat repeatedly in a lifetime… In fact, primitive warfare was much more deadly than that conducted between civilized states because of the greater frequency of combat and the more merciless way it was conducted’ (p.174).

Apart from political correctness, Keeley suggests that the importance of war in primitive societies has been underestimated because students of war have applied inappropriate ‘civilized’ criteria, and concluded wrongly that primitive war is ‘inefficient’. Pitched battles involving large numbers of fighters are rare in tribal warfare, and when they do occur they are sometimes of a semi-ritual nature, with few casualties. This has misled observers into thinking that war is not taken seriously in these societies. But according to Keeley the usual form of tribal war is continuous low intensity action - ambushes, raids on enemy villages, destruction of crops, etc. Male enemies are seldom taken prisoner, but killed on the spot (or taken for ritual torture or cannibalism), while females and children are taken as wives or slaves. ‘Primitive... warfare consists of war stripped to its essentials: the murder of enemies; the theft or destruction of their sustenance, wealth, and essential resources, and the inducement in them of insecurity and terror’ (p.75) Cumulatively the effects can be devastating, leading to tribes being annihilated or scattered and absorbed into other groups.

I can’t resist ending with Keeley’s description (p.99) of war in tribal Tahiti, which some 18th century explorers misguidedly saw as the embodiment of Rousseau’s happy state of nature:

“In Tahiti, a victorious warrior, given the opportunity, would pound his vanquished foe’s corpse flat with his heavy war club, cut a slit through the well-crushed victim, and don him as a trophy poncho”.